In a high frequency model i would like to add daily real interest rates (preferably short-term) as a variable.

However, I cannot seem to find either daily CPI data, or daily real interest rates (which go back to 1998). Is there some kind of daily proxy i could use, or maybe a way to calculate daily real rates with monthly data for CPI?

Since interest rate embeds forward expectations, a potentially better option is to use expected inflation (e.g., consensus economic forecasts). Depending on the data source, these could be of slightly higher frequency and are generally more timely. But in the end, some interpolations will still be needed.

The chart below shows two estimates, one using realized inflation and one with expected inflation:

I would NOT recommend using short-term real yield from TIPS, since these linkers are very illiquid and very technical in nature (with their yields reflecting a confluence of seasonality, illiquidity premium, and other idiosyncratic factors). The chart below shows 3-month TIPS yields over time: